1. Secants and Tangents A B T

2. A B A secant is a line that intersects a circle at exactly two points. (Every secant contains a chord of the circle.) T A tangent is a line that intersects a circle at exactly one point. This point is called the point of tangency or point of contact.

3. T T Postulate: A tangent line is perpendicular to the radius drawn to the point of contact. Postulate: If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle. Tangent lines.

4. Secant and Tangent Segments A tangent segment is a point of a tangent line between the point of contact and a point outside the circle. A secant segment is the part of a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle. T B Tangent segment A B RAR is the secant segment.

5. The external part of a secant segment is the part of a secant line that joins the outside point to the nearer intersection point. A B R BR is the external part.

6. Theorem 85: If two tangent segments are drawn to a circle from an external point, then those segments are congruent. (Two-Tangent Theorem) O A C B

7. Tangent Circles Tangent circles are circles that intersect each other at exactly one point. Two circles are externally tangent if each of the tangent circles lies outside the other. Two circles are internally tangent if one of the tangent circles lies inside the other. The point of contact lies on the line of centers. PQ PQ P Q

8. Common Tangents: PQ is the line of center XY is a common internal tangent. AB is a common external tangent. Definition: A common tangent is a line tangent to two circles (not necessarily at the same point.) Such a tangent is a common internal tangent if it lies between the circles(intersects the segment joining the centers) or a common external tangent if it is not between the circles (does not intersect the segment joining the centers.) A B Y P Q X

9. In practice, we will frequently refer to a segment as a common tangent if it lies on a common tangent and its endpoints are the tangent’s points of contact. In the diagram for example, XY can be called a common internal tangent and AB can be called a common external tangent. A B Y P Q X

10. Practice Draw five circles anywhere on your paper, label them 1-5. –Draw a tangent line to circle 1. –Draw a secant line to circle 2. –Draw a common tangent line between circles 3 and 4. –Draw an external tangent line between 4 and 5. –Draw an internal tangent line between 1 and 2.

11. 1.Draw the segment joining the centers. 2.Draw the radii to the points of contact. 3.Through the center of the smaller circle, draw a line parallel to the common tangent. 4.Observe that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right triangle. 5.Use the Pythagorean Theorem and properties of a rectangle.

12. Problem #1 A circle with a radius of 8 cm is externally tangent to a circle with a radius of 18 cm. Find the length of a common external tangent.

13. 1.Draw the segment joining the centers. 2.Draw the radii to the points of contact. 3.Through the center of the smaller circle, draw a line parallel to the common tangent. 4.Use the Pythagorean Theorem and properties of a rectangle to solve. 5.In ΔRPQ, (QR) 2 + (RP) 2 = (PQ) 2 1.10 2 + (RP) 2 = 26 2 2.RP = 24 6.AB = 24 cm 818 R

14. Problem #2 Given: Each side of quadrilateral ABCD is a tangent to the circle. AB = 10, BC = 15, AD = 18. Find CD. D C B A Let BE = x and “walk around” the figure using the given information and the Two- Tangent Theorem. E x 10 - x x 15 - x 10 - x 15 - x 18 – (10 – x ) CD = 15 – x + 18 – (10 – x) = 15 – x + 18 – 10 + x = 23

15. Finding the Lengths of Chords When two chords intersect in the interior of a circle, each chord is divided into two segments which are called segments of a chord. The following theorem gives a relationship between the lengths of the four segments that are formed.

16. Theorem 12.14 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. EA EB = EC ED

17. Proving Theorem 12.14 You can use similar triangles to prove Theorem 10.15. Given:, are chords that intersect at E. Prove: EA EB = EC ED

18. Proving Theorem 12.14 Paragraph proof: Draw and. Because  C and  B intercept the same arc,  C   B. Likewise,  A   D. By the AA Similarity Postulate, ∆AEC  ∆DEB. So the lengths of corresponding sides are proportional. = EA EB = EC ED Lengths of sides are proportional. Cross Product Property

19. Ex. 1: Finding Segment Lengths Chords ST and PQ intersect inside the circle. Find the value of x. RQ RP = RS RTUse Theorem 10.15 Substitute values.9 x = 3 6 9x = 18 x = 2 Simplify. Divide each side by 9.

20. 2x 15 10 12

21. Segments of Tangents and Secants In the figure shown, PS is called a tangent segment because it is tangent to the circle at an end point. Similarly, PR is a secant segment and PQ is the external segment of PR.

22. Theorem 12.16 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equal the square of the length of the tangent segment. (EA) 2 = EC ED

23. 10 6 x

24. 8 12

25. (BA) 2 = BC BDUse Theorem 10.17 Substitute values. 25 = x 2 + 4x 0 = x 2 + 4x - 25 Simplify. Use Quadratic Formula. (5) 2 = x (x + 4) Write in standard form. x =Simplify. Use the positive solution because lengths cannot be negative. So, x = -2 +  3.39. x =

26. Theorem 12.15 If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. EA EB = EC ED

27. Finding Segment Lengths Find the value of x. RP RQ = RS RTUse Theorem 10.16 Substitute values.9(11 + 9)=10(x + 10) 180 = 10x + 100 80 = 10x Simplify. Subtract 100 from each side. 8 = xDivide each side by 10.

28. 6 15 10 x

29. Ex. 3: Estimating the radius of a circle Aquarium Tank. You are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency is about 20 feet. Estimate the radius of the tank.

30. (CB) 2 = CE CDUse Theorem 10.17 Substitute values. 400  16r + 64 336  16r Simplify. 21  r Divide each side by 16. (20) 2  8 (2r + 8) Subtract 64 from each side.  So, the radius of the tank is about 21 feet.